Area, Perimeter & Pythagoras
We apply formulae for perimeter and area of polygons and circles, perform SI unit conversions, and solve problems involving composite figures. We also investigate how doubling a dimension affects area, and develop and apply the Theorem of Pythagoras to solve problems in right-angled triangles.
6.1 Perimeter and Area of Polygons
- Calculate the perimeter and area of squares, rectangles, triangles, and trapeziums
- Perform SI unit conversions for length and area
- Investigate the effect of doubling a dimension on area
Real-World Connection
Perimeter is the fence around your yard; area is the grass inside. A farmer needs perimeter to buy fencing and area to calculate how much seed to buy. A painter needs the area of each wall. Understanding the difference — and the effect of scaling — is crucial in construction, gardening, tiling, and manufacturing.
Perimeter and Area Formulae
Shape
Perimeter
Square (side s)
P = 4s
A = s²
Rectangle (l × w)
P = 2(l + w)
A = l × w
Triangle (base b, height h)
P = a + b + c
A = ½bh
Trapezium (parallel sides a, b; height h)
P = sum of sides
A = ½(a + b)h
SI Unit Conversions for Length
- 1 cm = 10 mm → 1 cm² = 100 mm²
- 1 m = 100 cm → 1 m² = 10 000 cm²
- 1 km = 1 000 m → 1 km² = 1 000 000 m²
⚠️ Warning
When converting AREA units, you must square the conversion factor. E.g. 1 m = 100 cm, so 1 m² = 100² cm² = 10 000 cm². Many students wrongly use ×100 instead of ×10 000.
Property / Rule
Effect of Scaling on Area
If you multiply ALL dimensions by a factor k, the area multiplies by k². Doubling all dimensions (k=2) gives 4× the area.
Worked Example
Calculating area of a composite figure
Problem
Worked Example
Perimeter and area of a trapezium
Problem
Worked Example
Effect of doubling dimensions on area
Problem
CAPS Cognitive Level Distribution
6.2 Circles — Circumference and Area
- Calculate the circumference and area of a circle
- Solve problems involving circles and composite figures containing circles
- Use the correct value of π (≈ 3.14 or π button on calculator)
Real-World Connection
Circles appear in wheels, pipes, coins, plates, and planets. The circumference of a tire determines how far a car travels per revolution. The area of a circular irrigation sprinkler determines how much land is watered. Pizza portion sizes are calculated using circular area formulae. π (pi) appears because the ratio of circumference to diameter is always the same for any circle.
Definition
π (Pi)
Pi is the ratio of any circle's circumference to its diameter. It is an irrational number: π ≈ 3.14159…
Circumference of a Circle
r = radius, d = diameter = 2r
Area of a Circle
r = radius; area is always in square units
💡 Tip
Remember: circumference uses r (not r²) and area uses r² (not r). A common error is swapping them. The units help: C has m (one dimension), A has m² (two dimensions).
ℹ️ Note
★ Extension — The third worked example (arc length and sector area) and the Level 4 activity go beyond the core CAPS Grade 9 bullets for this section. They are included for enrichment. Arc length and sector area are formally assessed from Grade 10 onwards.
Worked Example
Composite figure with a circle
Problem
Worked Example
Finding radius from area
Problem
Worked Example
Arc length and sector area
Problem
CAPS Cognitive Level Distribution
6.3 The Theorem of Pythagoras
- Investigate the relationship between the lengths of the sides of a right-angled triangle to develop the Theorem of Pythagoras
- Use the Theorem of Pythagoras to solve problems involving unknown sides in right-angled triangles
- Determine whether a triangle is right-angled given the lengths of its three sides (converse of Pythagoras)
- Apply the Theorem of Pythagoras in two-step problems and real-world contexts
Real-World Connection
Ancient Egyptians used a rope with 12 equally spaced knots to form a 3-4-5 right triangle and lay perfectly square corners for the pyramids. Today, builders use the same idea — a spirit level tells you the angle is 90°, but Pythagoras tells you the exact diagonal length. Carpenters, architects, surveyors, and GPS systems all rely on this theorem every day.
Definition
Right-angled triangle
A triangle with one angle equal to 90°. The side opposite the right angle is always the longest side and is called the hypotenuse. The other two sides are called the legs (adjacent sides).
Theorem of Pythagoras
c = hypotenuse (longest side, opposite the right angle); a and b are the two legs. The theorem states: the square of the hypotenuse equals the sum of the squares of the other two sides.
📜 Did You Know?
The theorem was known to Babylonians over 3 700 years ago and to ancient Indians and Chinese mathematicians. Pythagoras of Samos (c. 569–475 BC) is credited with the first formal mathematical proof. It is used in algebra, trigonometry, measurement, analytical geometry, and GPS navigation.
Property / Rule
Finding the hypotenuse
When you need the hypotenuse (c), square both legs, add, then take the square root.
Property / Rule
Finding a leg
When the hypotenuse and one leg are known, subtract the known leg squared from the hypotenuse squared, then take the square root.
Property / Rule
Converse of Pythagoras
If the three sides of a triangle satisfy a² + b² = c², where c is the longest side, then the triangle MUST be right-angled. Use this to TEST whether a triangle has a 90° angle.
Common Pythagorean triples (integer sides)
- 3 ; 4 ; 5 — check: 9 + 16 = 25 ✓ (multiply by any factor: 6-8-10, 9-12-15, etc.)
- 5 ; 12 ; 13 — check: 25 + 144 = 169 ✓
- 8 ; 15 ; 17 — check: 64 + 225 = 289 ✓
- 7 ; 24 ; 25 — check: 49 + 576 = 625 ✓
🚨 Common Mistake
Always identify the hypotenuse FIRST — it is opposite the right angle (90°), NOT adjacent to it. The hypotenuse is ALWAYS the longest side. Many errors come from squaring the wrong side.
Worked Example
Finding the hypotenuse
Problem
Worked Example
Finding a leg
Problem
Worked Example
Testing the converse
Problem
CAPS Cognitive Level Distribution